All continuous maps of an arbitrary space $X$ into the segment $I$ are homotopic to each other. In Lecture 3 of Fuchs-Fomenko's "Homotopical Topology", it says the following:
All continous maps of an arbitrary space $X$ into the segment $I$ are homotopic to each other: A homotopy $h_t: X \to I$ joining continuous maps $f, g: X \to I$ is defined by the formula $h_t(x) = (1-t)f(x) + tg(x)$. Here $I$ can be replaced by any convex subset of any space $\mathbb{R}^n$ or $\mathbb{R}^\infty$, in particular, by the whole spaces $\mathbb{R}^n$ and $\mathbb{R}^\infty$.
My question is this: I can see that each map $h_t$ is continuous. But why is the map $H: X \times I \to I$ given by $H(x, t) = h_t$ is continuous? Thanks.
 A: Writing $H(x,t) = (1-t) f(x) + t g(x)$, one simply applies continuity theorems, one after the other, as follows:

*

*The projection maps $p(x,t) =x$ and $q(x,t)=t$ are continuous;

*The compositions $f(p(x,t))=f(x)$ and $g(q(x,t))=g(t)$ of continuous functions are continuous;

*The constant function $c(x,t)=1$ is continuous;

*The difference $1-t=c(x,t)-q(x,t)$ of continuous functions is continuous;

*The products $(1-t) f(x)$ and $t g(x)$ of continuous functions are continuous;

*And, finally, the sum $H(x,t) = (1-t)f(x) + t g(x)$ of continuous functions is continuous.

One thing to keep in mind: $H$ is a function whose domain is the product space $X \times I$, and one has a theorem in that situation, using the product topology, which says that each of the two projection functions $p(x,t)=x$ and $q(x,t)=t$ are continuous. Intuitively, this allows you to throw $x$'s and $t$'s around in your formula however you like, and as long as they are being combined in a continuous fashion, you'll get a continuous function.
A: That map is continuous because it's a sum of products of continuous functions, so it is continuous by basic point-set topology/analysis.
Moreover,  if you can replace $I$ by any convex subset of any Euclidean space, why not a point? Then all maps are constant, which are not just homotopic but equal. Then the homotopy between your maps $f,g$ is obtained by composing $f$ with the homotopy equivalence between $I$ and a point, and back again for $g$.
